doi:10.1016/j.asoc.2004.09.002 
Copyright © 2004 Elsevier B.V. All rights reserved.
Efficient decomposition methods of fuzzy relation and their application to image decomposition
aDepartment of Computational Intelligence and Systems Science, Tokyo Institute of Technology, 4259 Nagatsuta, Midiri-ku, Yokohama 226-8502, Japan
bDICOMMA, University of Napoli, “Federico II”, Via Monteoliveto 3, 80134 Napoli, Italy
cDepartment of Electrical and Computer Engineering, University of Alberta, Edmonton T6R 2G7, Canada
Received 30 April 2003;
revised 2 July 2004;
accepted 8 September 2004.
Available online 26 November 2004.
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Abstract
Two optimizations for decomposition problem of fuzzy relation (image) are proposed. The first optimization is a fast decomposition method of fuzzy relation based on the properties of max and min operations and the simultaneous updating of the prototype. The second optimization corresponds to an improvement of a cost function, in order to obtain a good quality of the solution of the decomposition problem.
Keywords: Fuzzy relation; Optimization; Gradient method; Image decomposition
Fig. 1. Overview of decomposition of fuzzy relation.
Fig. 2. Original fuzzy relation (No. 611000, Arizona directory, Corel Gallery, left), initial random pattern (right).
Fig. 3. Overview of proposed method.
Fig. 4. Approximated fuzzy relation c = 50 (left, Q = 1224.57) and c = 100 (right, Q = 893.23).
Fig. 5. Decomposed fuzzy sets, A (left) and B (right), Schein rank = 50.
Fig. 6. Decomposed fuzzy sets, A (left) and B (right), Schein rank = 100.
Fig. 7. Original fuzzy relation “text” (left), approximate fuzzy relation (c = 50, middle, Q = 1500.41), (c = 100, right, Q = 1261.63).
Fig. 8. Decomposed fuzzy sets, A (left) and B (right), Schein rank = 50.
Fig. 9. Decomposed fuzzy sets, A (left) and B (right), Schein rank = 100.
Fig. 10. Original fuzzy relation “texture” (left), approximate fuzzy relation (c = 50, middle, Q = 381.48), (c = 100, right, Q = 351.14).
Fig. 11. Decomposed fuzzy sets, A (left) and B (right), Schein rank = 50.
Fig. 12. Decomposed fuzzy sets, A (left) and B (right), Schein rank = 100.
Fig. 13. Local window on approximated image.
Fig. 14. Comparison of approximated images, Schein rank = 50, α = 0.005, left (β = 0), right (β = 0.3).
Fig. 15. Comparison of approximated images, Schein rank = 100, α = 0.01, left (β = 0), right (β = 0.05).
Fig. 16. Comparison of approximated images, Schein rank = 200, α = 0.01, left (β = 0), right (β = 0.1).
Fig. 17. SQE with respect to iteration number (Schein rank = 50, α = 0.005).
Fig. 18. SQE with respect to iteration number (Schein rank = 100, α = 0.01).
Fig. 19. SQE with respect to iteration number (Schein rank = 200, α = 0.1).
Table 1.
Computation time comparisons
Table 2.
Index Q and iteration number comparison
Table 3.
Comparison of SQE and iteration number (rank = 50, w = 3)
Table 4.
Comparison of SQE and iteration number (rank = 100, w = 3)
Table 5.
Comparison of SQE and iteration number (rank = 200, w = 3)
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